# Complex equations

**URL:** https://discourse.julialang.org/t/complex-equations/94742
**Category:** New to Julia
**Tags:** modelingtoolkit
**Created:** [February 16, 2023, 8:29pm UTC](https://discourse.julialang.org/t/complex-equations/94742 "2023-02-16T20:29:06Z")
**Posts on this page:** 2
**Page:** 1

<div class="post-metadata">

### Author: ![johhell](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/johhell/32/46165_2.png) [@johhell](https://discourse.julialang.org/u/johhell)
#### Post date: [February 16, 2023, 8:29pm UTC](https://discourse.julialang.org/t/complex-equations/94742/1 "2023-02-16T20:29:06Z")

</div>

After \>20 years working with MODELICA I tried to use Complex equations in `MTK`.

In a simple example an ohmic resistor and a inductor are connected in series to a constant voltage source. I used per unit system with space-phasors and variable speed of reference frame `@parameters wReference`.

```julia
using ModelingToolkit
using DifferentialEquations
using Plots

frequency = 50.0
Omegarated = frequency*2pi
JJ = 1.0im

function NewVars(eqName::String, CCX::Vector{Complex{Num}})::Tuple{Expr, Expr}
    vari::String = "newVariables = @variables "
    substi::String = ""
    for cx in CCX
        stcx = string(cx)[1:end-3] # without '(t)'
        vari = string(vari,stcx*"₋re(t) ",stcx*"₋im(t) ")
        substi = string(substi,"$eqName = substitute($eqName,(real($stcx)=>$(stcx)₋re)); ")
        substi = string(substi,"$eqName = substitute($eqName,(imag($stcx)=>$(stcx)₋im)); ")
    end
    return (Meta.parse(vari), Meta.parse(substi))
end

@variables t rad(t)=0.0 w(t)

CX = @variables begin
    deri(t)::ComplexF64
    amps(t)::ComplexF64
    gridVolts(t)::ComplexF64
end

@parameters x r wReference = 1.0

D = Differential(t)
# DiffC(cpx) = Complex(D(real(cpx)),D(imag(cpx))) # used for testing

#NOTE
#NOTE this line was added in Symbolics - diff.jl
# (D::Differential)(x::Complex{Num}) = Num(D(real(x)))+Num(D(imag(x)))*im
#NOTE

eqs1 = [ D(amps) ~ deri*Omegarated
        gridVolts ~ x*(deri + wReference*amps*JJ) + r*amps
        w ~ D(rad)
        w ~ Omegarated*(1.0-wReference)
        gridVolts ~ Complex(cos(rad), sin(rad))
    ]

eqs=deepcopy(eqs1)

(Bvaria, Bsubst) = NewVars("eqs",CX)
eval(Bvaria) # create new variables
eval(Bsubst) # substitute

pr = 0.3; px = 0.1
u0 = [0.0, 0.0, 0.0]
p = [x => px, r=> pr, wReference => 1.0]

# exact solution
strom = (1.0+0im)/(pr+JJ*px)
println("exact solution: amps = $(strom)")

@named model = ODESystem(eqs)
sys = structural_simplify(model)

prob = ODEProblem(sys, u0, (0., 1.0/frequency), p)
sol = solve(prob, Tsit5())
plot(sol)

```

## my approach

### (D::Differential)(x::Complex{Num}

an additional line in `diff.jl`  
`(D::Differential)(x::Complex{Num}) = Num(D(real(x)))+Num(D(imag(x)))*im`

### substitute real(x), imag(x)

`real(x), imag(x)` are replaced by new @variables x\_re x\_im

## Questions

- This works with monolithic models. But how to do with component based models? (e.g. `connect()` and `Flow` )

### backup alternative

To split Complex equations manually. This works, but `Symbolics` should do this for me.

Thanks

---

<div class="post-metadata">

### Author: ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [February 16, 2023, 9:39pm UTC](https://discourse.julialang.org/t/complex-equations/94742/2 "2023-02-16T21:39:39Z")

</div>

> [@johhell](#):
>
> To split Complex equations manually. This works, but `Symbolics` should do this for me.

It should, that’s just not all complete yet. For now the splitting needs to be done manually, but it’s on the menu.